Basic Math Examples

$\frac{\sqrt[3]{9}}{\sqrt[5]{27}}$

Step 1

Multiply

$\frac{\sqrt[3]{9}}{\sqrt[5]{27}}$ by

$\frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$ .

$\frac{\sqrt[3]{9}}{\sqrt[5]{27}} \cdot \frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply

$\frac{\sqrt[3]{9}}{\sqrt[5]{27}}$ by

$\frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{\sqrt[5]{27} {\sqrt[5]{27}}^{4}}$

Step 2.2

Raise

$\sqrt[5]{27}$ to the power of

$1$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{1} {\sqrt[5]{27}}^{4}}$

Step 2.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{1 + 4}}$

Step 2.4

Add

$1$ and

$4$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{5}}$

Step 2.5

Rewrite

${\sqrt[5]{27}}^{5}$ as

$27$ .

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Step 2.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[5]{27}$ as

$27^{\frac{1}{5}}$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{(27^{\frac{1}{5}})}^{5}}$

Step 2.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{\frac{1}{5} \cdot 5}}$

Step 2.5.3

Combine

$\frac{1}{5}$ and

$5$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{\frac{5}{5}}}$

Step 2.5.4

Cancel the common factor of

$5$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{\frac{5}{5}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{1}}$

Step 2.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite

${\sqrt[5]{27}}^{4}$ as

$\sqrt[5]{27^{4}}$ .

$\frac{\sqrt[3]{9} \sqrt[5]{27^{4}}}{27}$

Step 3.2

Raise

$27$ to the power of

$4$ .

$\frac{\sqrt[3]{9} \sqrt[5]{531441}}{27}$

Step 3.3

Rewrite

$531441$ as

$9^{5} \cdot 9$ .

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Step 3.3.1

Factor

$59049$ out of

$531441$ .

$\frac{\sqrt[3]{9} \sqrt[5]{59049 (9)}}{27}$

Step 3.3.2

Rewrite

$59049$ as

$9^{5}$ .

$\frac{\sqrt[3]{9} \sqrt[5]{9^{5} \cdot 9}}{27}$

Step 3.4

Pull terms out from under the radical.

$\frac{\sqrt[3]{9} \cdot 9 \sqrt[5]{9}}{27}$

Step 3.5

Combine exponents.

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Step 3.5.1

Rewrite the expression using the least common index of

$15$ .

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Step 3.5.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{9}$ as

$9^{\frac{1}{3}}$ .

$\frac{9 (9^{\frac{1}{3}} \sqrt[5]{9})}{27}$

Step 3.5.1.2

Rewrite

$9^{\frac{1}{3}}$ as

$9^{\frac{5}{15}}$ .

$\frac{9 (9^{\frac{5}{15}} \sqrt[5]{9})}{27}$

Step 3.5.1.3

Rewrite

$9^{\frac{5}{15}}$ as

$\sqrt[15]{9^{5}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \sqrt[5]{9})}{27}$

Step 3.5.1.4

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[5]{9}$ as

$9^{\frac{1}{5}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \cdot 9^{\frac{1}{5}})}{27}$

Step 3.5.1.5

Rewrite

$9^{\frac{1}{5}}$ as

$9^{\frac{3}{15}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \cdot 9^{\frac{3}{15}})}{27}$

Step 3.5.1.6

Rewrite

$9^{\frac{3}{15}}$ as

$\sqrt[15]{9^{3}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \sqrt[15]{9^{3}})}{27}$

Step 3.5.2

Combine using the product rule for radicals.

$\frac{9 \sqrt[15]{9^{5} \cdot 9^{3}}}{27}$

Step 3.5.3

Multiply

$9^{5}$ by

$9^{3}$ by adding the exponents.

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Step 3.5.3.1

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 \sqrt[15]{9^{5 + 3}}}{27}$

Step 3.5.3.2

Add

$5$ and

$3$ .

$\frac{9 \sqrt[15]{9^{8}}}{27}$

Step 3.6

Raise

$9$ to the power of

$8$ .

$\frac{9 \sqrt[15]{43046721}}{27}$

Step 3.7

Rewrite

$43046721$ as

$3^{15} \cdot 3$ .

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Step 3.7.1

Factor

$14348907$ out of

$43046721$ .

$\frac{9 \sqrt[15]{14348907 (3)}}{27}$

Step 3.7.2

Rewrite

$14348907$ as

$3^{15}$ .

$\frac{9 \sqrt[15]{3^{15} \cdot 3}}{27}$

Step 3.8

Pull terms out from under the radical.

$\frac{9 (3 \sqrt[15]{3})}{27}$

Step 4

Multiply

$9$ by

$3$ .

$\frac{27 \sqrt[15]{3}}{27}$

Step 5

Cancel the common factor of

$27$ .

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Step 5.1

Cancel the common factor.

$\frac{27 \sqrt[15]{3}}{27}$

Step 5.2

Divide

$\sqrt[15]{3}$ by

$1$ .

$\sqrt[15]{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[15]{3}$

Decimal Form:

$1.07598962 \dots$